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Questions tagged [calculus-of-variations]

Optimization of functionals mostly defined on infinite-dimensional spaces.

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Gamma ($\Gamma$) Convergence of Functionals

Consider a set $X$, and consider a sequence of functionals on $X$, that is maps $F_n: X \to \mathbb{R}$. We say that $F_n$ "$\Gamma$" converges to $F$, if the limit satisfies: $F(x) \leq \inf\{\...
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Euler-Lagrange equation alternative form

I have the following exercise: Let $a,b,A,B\in \mathbb{R},a<b,f\in C^2\left(\left[a,b\right]\times \mathbb{R}\right),$ and $J:\mathcal{M}\rightarrow\mathbb{R}$ given by $$J\left(y\right)=\int_a^bf\...
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1answer
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Classical Lagrangian invariant under transformation

Consider the Lagrangian $$L(q_1, q_2, \dot{q_1}, \dot{q_2}) = \dot{q_1}^2 - \dot{q_2}^2 + q_1 ^2 - q_2 ^2$$ (Set aside any concerns about the possibility of the kinetic energy being negative.) Show ...
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Lagrange Multipliers with Three constraints .

Let $J$ be the Functional $ J:X \rightarrow \mathbb{R}$ where $X$ is a Banach space of functions, I would Like to minimize this functional under three constraint $F_1,F_2$ and $F_3$ such that $ F_i:...
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A reverse of Lax-Milgram Theorem

Let $V$ be a Hilbert space. Let $a:V\times V\rightarrow\mathbb{R}$ be a symmetric bilinear continuous mapping such that $a(v,v)\geq 0$ for any $v\in V$. Assume that for any continuous linear ...
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Why can the Euler-Lagrange equation be used to find the extremum of a functional?

The Euler-Lagrange equation is a differential equation. A functional is a function from some vector space to a real number. Functions that maximize or minimize functionals may be found using the Euler–...
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Calculus of variation: quasi convex function

I am given the following things: let $M\in \mathbb{R}^{(N\times n)\times(N\times n)}$ be a symmetric matrix. The function $f: \mathbb{R}^{N\times n} \to \mathbb{R}$ is defined as: $$f(\xi):=\langle M\...
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f is convex if and only if $f(x):=\left \langle Mx,x\right \rangle \ge 0$

I have to show for $M \in \mathbb{R}^{(N \times n)\times (N \times n)}$ symmetric Matrix and $f:\mathbb{R}^{N\times n} \longrightarrow \mathbb{R}$ with $f(x):= \langle Mx,x\rangle$ that: f is convex ...
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14 views

$\Gamma$ convergence of $W^{1,p_n}$ norm as $p_n\rightarrow p$

Given $\Omega$ a bounded and open subset of $\mathbb R^N$. And a sequence $p_n$ such that $p_n>p_0>1$ and $p_n\rightarrow p$. Define the functional $$ F_n(u)=||u||_{W_{\varphi}^{1,p_n}(\Omega)}, ...
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Derivative of a special double integrale in the calculus of variation

Let $u$ be a real function defined on $[0, T]$ and the functional $$V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy$$ and $f(x,y)$ is symmetric , continuous and can be writen on this ...
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Calculus of Variations Boundary Terms in Higher Dimensions

In 2D I apply the calculus of variations to get an equation of the following form on a domain $\Omega\subset\mathbb{R}^2$, and I want to extract the Euler-Lagrange equations: $$\int_{\Omega} f(p)\...
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Prove that there exists a function $ g: [0,1] \to \mathbb{R} $ which is 1-Lipschitz, satisfies $ g (0) = 0 $ and has the following property.

Question. Consider a continuous function $ f: [0,1] \to \mathbb {R} $. Prove that there exists a function $ g: [0,1] \to \mathbb{R} $ which is 1-Lipschitz, satisfies $ g (0) = 0 $ and has the property ...
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Find function in $W^{1,1}(0,1)$

I want to show that there is no minimizer for $F[u]:=\int \limits_{0}^{1} \sqrt{u(x)^2+u^{\prime}(x)^2}dx$ in $\mathcal{Z}:= \lbrace u \in W^{1,1}(0,1):u(0)=0 \; \; \text{and} \; \; u(1)=1 \rbrace $. ...
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Morrey growth condition implies Hölder continuity

It is know that the growth condition $$\sup_{x \in \Omega }\int_{B_r(x)\cap \Omega} |\nabla u|^2 \leq C r^{n-2+2\alpha} \text{ for all } r>0$$ would imply that $u \in C^{0, \alpha}(\Omega)$. I ...
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Condition for curve to be an extremal does not depend on the choice of coordinate system

I am reading V.Arnold book "Mathematical Methods of Classical Mechanics" and on page 59 he mentions that "the condition for a curve $\gamma$ to be an extremal of a functional does not depend on the ...
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Inverse functional derivatives: Find a functional whose functional derivative is a given function

https://en.wikipedia.org/wiki/Functional_derivative Is there a straightforward way to find a functional whose functional derivative has a particular form? That is, is there something like functional ...
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1answer
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Prove the function has infinitely many local maxima and no local minima

I need to find the local minima and maxima of f $$f(x,y)=(1+e^y)\cos{x}-ye^y.$$ I have found that $\nabla{f}(2k\pi,0)=0$ and $\nabla{f}\big((2k+1)\pi,-2\big)=0,\forall{k\in{\mathbb{Z}}}$. How can I ...
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1answer
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Integration by Parts in Two Dimensions - Plate Theory

I am trying to evaluate the variation of strain energy for a thin plate to obtain the correct form of the boundary conditions associated with the problem. The strain energy, $U$ is given in terms of ...
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Showing that a transformation is a variational symmetry

I'm trying to solve problem 9.2.1 in the book 'The calculus of variations' by Bruce Van Brunt. I was given the functional $$J(y)=\int_{x_0}^{x_1}xy'^2 dx$$ Now I'm supposed to show that the ...
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1answer
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Why is $f_\epsilon(u) \in W_0^{1,2}(\Omega)$?

For $\epsilon>0$ let $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$ One calculates that $\nabla f_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u $ , for $\epsilon$ to 0 this term goes to $\...
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Variational method in classical mechanic's problem

Consider the problem $$ (*)\begin{cases} x''(t)= F(x(t)) \\x(0)=P, x(1)=Q \end{cases}$$ where $P,Q\in\mathbb{R}^3$, $x=(x_1,x_2,x_3)$ and $F=-\nabla U$ for some potential $U:\mathbb{R}^3\rightarrow\...
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1answer
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Why is $f_\epsilon(u) \in H_0^{1,2}(\Omega)$?

I have to proof that $ u \in H_0^{1,2}(\Omega)$ implies $|u| \in H_0^{1,2}(\Omega)$ . I can define $f_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$ . it follows that $\nabla f_\epsilon(u)=\frac{u}{\...
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1answer
32 views

Weakly lower semicontinuity involving cosine

I want to stablish the existence of $T$-periodic weak solution to $u''+\sin (u)=h$, where $h$ is continuous and $T$-periodic on $\mathbb{R}$ with zero mean. In order to do that I want to find minimum ...
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1answer
23 views

Find adjoint operator defined on element

Let $E = L^2 (0, 1)$. Given $u ∈ E$, set $Tu(x)=\int_0^x u(t)dt$. Find $T^*$. Solution says only $(u, T^* v) = \int_t^1 v(x)dx$. I dont understand. adjoint operator is defined by $(Tu,v)=(u,T^*v)$. ...
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What are the use cases of the Dirichlet energy in computer vision?

I am reading a paper, in the context of computer vision, that mentions the "famous" Dirichlet energy. I am not familiar with this Dirichlet energy, but apparently we can minimise it. What are specific ...
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An Implication in du Bois-Reymond Lemma of Dimension 2?

Assume that $ g = g(x,y) \in L^1((0,2\pi)\times (0,2\pi)) $ and satifies \begin{equation*} \int^{2\pi}_0 \alpha'(x) g(x,y) dx = A(\alpha), \ \forall \alpha \in C^\infty_0(0,2\pi), \end{equation*} ...
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1answer
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How to Solve a Calculus of Variation Problem with Terminal Conditions

I've been struggling to solve calculus of variation problems with terminal conditions. The current textbook i'm using for my course seems to only tangentially touch upon the methodology. Here is one ...
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Two proofs of the fundamental theorem of calculus of variations - one correct, one not?

Fundamental Theorem of the Calculus of Variations. Let $u \in L^1_{\text{loc}}(a,b)$ and $$ \int_{a}^{b} u(x) \varphi(x) dx = 0 \quad \forall \varphi \in \mathcal{C}^{\infty}_{\text{c}}(a,b). $$ ...
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1answer
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Why can't we derive Pontryagin's adjoint method by the chain rule?

I'm trying to read the supplement to the NeurIPS 2018 best paper winner, Neural Ordinary Differential Equations, and am having trouble understanding the derivation of Pontryagin's adjoint method. So ...
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Find the extremal of the given function. CSIR -DEC 2018

$j[y]=\int_{0}^{1}[(y')^{2}-(y')^{4}]dx$ ,subject to condition $y(0)=0,y(1)=0.$A broken extremal is a continuous extremal whose derivative has jump discontinuities at a finite number of points.Then ...
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How to find a function having following properties?

Let the function be $f(x)$ such that it is monotonic and continuous function It has maximum area between $x=0$ and $x=1$ $f'(x)$ = $f(x)(1-f(x))$ $f''(0.5) =0$ thus, gradient is maximum at 0.5. ...
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prove unicity of a linear continous form

Let H be a Hilbert space and $ u ∈ H$. Prove there exists a unique continuous linear form $L∈ H^*$ such that: $||L||_{H^∗} = ||u||_H$ and $<L, u> = ||u||^2_H$ I proved the existence : We can ...
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Optimal control problem - Hamiltonian won't work

Let the following optimal control problem $(r>0)$: $$\max \int_0^T\ln q(t)\exp(-rt)dt$$ $$x(0)=s \quad \text{and} \quad x'(t)=-q(t)$$ My idea : let $\lambda \in C^1[0,T]$ then $\mathcal{H}(t,x,...
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Euler Equation in Optimal Control Problem

Let $c$ a continuous convex function. And we consider the following problem $$\int_0^1[x(t)E(t)-c(E(t))]dt$$ $$x(0)=1000 \quad x(1)=500$$ $$x'(t)=-x(t)E(t)$$ the exercise asks for Euler-Lagrange ...
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How to get this boundary condition?

$\int_{\partial \Omega}(-v\nabla \Delta u\cdot\mathbf{n}+\alpha\nabla^{T}vH_{u} \mathbf{n})ds=0$, for all $v$ which satisfies$\nabla v\cdot n =0$ on $\partial \Omega$, derive the boundary condition:$\...
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1answer
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Computing projection onto the following closed convex set

Let $\mathbf{S}^n$ denote the space of symmetric, real-valued $n \times n$ matrices. Consider the closed convex set $$ \mathcal{C} := \{(X, x) \in \mathbf{S}^n \times \mathbf{R}^n : X \succeq xx^T,...
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Optimal trajectory under acceleration constraint

Let $x(t) \in \mathbb{R}$, such that $x(0) = x_0$, $\dot{x}(0) = v_0$ Find the optimal trajectory for $x(t)$, such that $$L = \int_{t=0}^T|x(t) - x_f|^2dt$$ is minimized, subject to the constraint $|\...
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A functional is bounded on a bounded subset of Hilbert Space

Let $H$ be a Hilbert Space with the norm $||\,\cdot\,||$ and inner product $(\, ,\,)$. Define $I : H\to \mathbb{R}$ as a nonlinear functional on $H$. Definition 1(Differentiability of I) $I$ is ...
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Show that a functional has no extremal

I'm studying variations of calculus and it is quite new for me and I had problem with one exercise in my book. I was given a functional $$J(y)=\int_{-1}^{1}x^4\big(y'\big)^2 dx $$ and I'm supposed to ...
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Confusion on minimizing an integral function subject to integral constraint

I am trying to minimize the function $$\int_{0}^{1} y(t)\sqrt{1+(y'(t))^2} dt$$ under the constraints that $y(0)=y(1)=0$ and $\int_{0}^{1}\sqrt{1+(y'(t))^2} dt=2$. My intuition is can you not just ...
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Deriving Euler-Lagrange equation from minimization problem

Let $J[u]=\int_{x_1}^{x_2}g(x,u(x))\sqrt{1+(\frac{du}{dx})^2}dx$ where $g(x,u(x))$ is a smooth function. The following minimization problem is given: Find $u(x) $ subject to $u(x_1)=f_1$ and $u(x_2)=...
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Calculus of variation problem doesn't have a smooth solution

we define the minimization problem : $$\min J(x)=\int_{-1}^1x^2(t)[x'(t)-1]dt \quad \text{subject to }x(-1)=0 \quad \text{and} \quad x(1)=0$$ note that $0\leq J(x)$. and if we consider $x^*(t)=0 \...
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Proving that two variational problems are equivalent

Let $\Omega$ be an open set of finite measure. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue for the Laplace operator, i.e. $$- \Delta u = \lambda_1(\Omega) u, \ \ \ \ \ \text{in} \ \...
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Taking variational vs partial derivative Lagrange Multipliers

So I'm trying to solve the optimization problem shown in the picture below. The functional then becomes $$P = \int_0^L \frac12 E A \epsilon^2-pu+\sigma \left( \frac{du}{dx}-\epsilon \right)dx = \text{...
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Can the condition of this theorem be made weaker?

The proof seems fine to me, but I wonder if we could replace all the $F_{y'}$ with $F$ so that we can loose the condition from continuous second derivatives to continuous first derivative?
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Convergence of Frechet Derivative Functional related to Palais-Smale Sequence

Let $\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$ be an energy functional for bounded domain $\Omega$ such that $\phi'$ be its Frechet derivative, and there exists a bounded sequence $\{u_{n}\}_{n\in\mathbb{...
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1answer
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Prove continuity in the $p$-mean / Lebesgue Points

Lemma: Let $p \in [1, \infty)$, $I := (a,b)$ a real interval and $u \in L^p(I)$ arbitrary. Then we have \begin{equation*} \forall \varepsilon > 0 \ \exists \delta > 0: | h | < \...
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1answer
25 views

Reference Request: Original $\Gamma$-convergence Paper

Does anyone know what the original paper/work of Ennio de Giorgi, where $\Gamma$-convergence first appeared is?
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Young measures, concentration and oscillation

Given the generic example for motiving Young-measures, namely minimising the functional $$ \mathcal{F}(u)= \int_0^1 u(x)^2 + (u'(x)^2-1)^2 \mathrm{d}x, \quad u(0)=u(1)=0,$$ motivates the ...
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Frechet Derivative and Convergence of Functionals

Let $\Omega\subset\mathbb{R}$ be a bounded interval, $\{u_{n}(t)\}_{n\in\mathbb{N}}$ be a bounded sequence in $H_{0}^{1}(\Omega)$ and define $J[u_{n}(t)] = \frac{1}{2}||u_{n}(t)||_{H_{0}^{1}(\Omega)}^{...